通过双向协作学习，从不完整观测中联合恢复PDE参数与状态
Bi-HYCO: Bi-Objective Cooperative Learning for PDE Parameter Identification under Fragmented Observations
- 构建双目标协同框架，共享未标记交互点的状态预测
- 在椭圆传输与二维纳维-斯托克斯实验中实现高精度参数重建
- 适合处理多源观测不完整且需联合优化的物理建模场景
物理模型与合成模型可能描述同一类由偏微分方程（PDE）控制系统的互补方面，但接收不同的、可能不完整的观测数据。我们提出双目标协同学习（Bi-HYCO），该框架保留两种表示及其局部观测目标，并在无测量的未标记交互点上耦合其预测状态。这些点不扩充数据量，仅在公共状态空间中提供通信机制。两个目标构成向量值目标函数，加权标量化提供计算实现。针对具有固定交互点的确定性共享观测算法，我们证明了整个交替序列的充分下降性和有限长度，且在所提出的Kurdyka-Łojasiewicz型假设下收敛至混合临界点。通过椭圆传输和二维纳维-斯托克斯实验评估参数与状态重构效果、噪声及标化方式的影响，并与PINN/XPINN方法对比。消融实验表明，即使保留聚合机制，移除状态交互也会显著恶化参数恢复性能，尤其在纳维-斯托克斯案例中。
原文摘要 · Abstract (English)
Physical and synthetic models may describe complementary aspects of the same PDE-governed system while receiving different, possibly fragmented, observations. We propose Bi-Objective HYCO (Bi-HYCO), a cooperative framework that retains both representations and their local observational objectives while coupling their predicted states at unlabeled interaction points. These points contain no measurements and do not augment the data; they provide a communication mechanism in the common state space. The two criteria form a vector-valued objective, and weighted scalarizations provide computational realizations. For the deterministic shared-observation algorithm with fixed interaction points, we prove sufficient decrease and finite length of the whole alternating sequence, which converges to a mixed critical point under the stated Kurdyka-Lojasiewicz-type assumptions. Elliptic transmission and two-dimensional Navier-Stokes experiments assess parameter and state reconstruction, noise and scalarization effects, and PINN/XPINN references. Ablations show that removing state interaction while retaining aggregation deteriorates parameter recovery in the tested configurations, particularly for Navier-Stokes.
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